Dimension of harmonic measures in hyperbolic spaces

TL;DR

This paper proves the exact dimension of harmonic measures for random walks on hyperbolic groups, providing a formula relating entropy and drift, with applications to Hausdorff dimension continuity and Brownian motions.

Contribution

It establishes a dimension formula for harmonic measures in hyperbolic spaces, including non-proper cases, under finite first moment conditions, extending previous results.

Findings

Dimension formula as entropy over drift

Continuity of Hausdorff dimension with respect to measures

Application to Brownian motions on Riemannian coverings

Abstract

We show exact dimensionality of harmonic measures associated with random walks on groups acting on a hyperbolic space under finite first moment condition, and establish the dimension formula by the entropy over the drift. We also treat the case when a group acts on a non-proper hyperbolic space acylindrically. Applications of this formula include continuity of the Hausdorff dimension with respect to driving measures and Brownian motions on regular coverings of a finite volume Riemannian manifold.